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On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting
Hypocoercivity of linear kinetic equations via Harris's Theorem
1. | Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain |
2. | CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris, France |
3. | BCAM Basque Center for Applied Mathematics, Alameda Mazarredo, 14, 48009 Bilbao, Spain |
We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $ (x,v) \in \mathbb{T}^d \times \mathbb{R}^d $ or on the whole space $ (x,v) \in \mathbb{R}^d \times \mathbb{R}^d $ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $ L^1 $ or weighted $ L^1 $ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.
References:
[1] |
D. Bakry, P. Cattiaux and A. Guillin,
Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.
doi: 10.1016/j.jfa.2007.11.002. |
[2] |
V. Bansaye, B. Cloez and P. Gabriel,
Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions, Acta Applicandae Mathematicae, (2017), 1-44.
doi: 10.1007/s10440-019-00253-5. |
[3] |
A. Bensousan, J. L. Lions and G. C. Papanicolau,
Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
[4] |
P. G. Bergman and J. L. Lebowitz,
New approach to nonequilibrium processes, Phys. Rev. (2), 99 (1955), 578-587.
doi: 10.1103/PhysRev.99.578. |
[5] |
E. Bernard and F. Salvarani,
On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954.
doi: 10.1016/j.jfa.2013.06.012. |
[6] |
M. Bisi, J. A. Cañizo and B. Lods,
Entropy dissipation estimates for the linear Boltzmann operator, J. Funct. Anal., 269 (2015), 1028-1069.
doi: 10.1016/j.jfa.2015.05.002. |
[7] |
M. Briant,
Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.
doi: 10.3934/krm.2015.8.281. |
[8] |
M. Briant,
Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.
doi: 10.1007/s00205-015-0874-x. |
[9] |
J. A. Cañizo, J. A. Carrillo and S. Cuadrado,
Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.
doi: 10.1007/s10440-012-9758-3. |
[10] |
J. A. Cañizo, A. Einav and B. Lods,
On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials, J. Math. Anal. Appl., 462 (2018), 801-839.
doi: 10.1016/j.jmaa.2017.12.052. |
[11] |
M. J. Cáceres, J. A. Carrillo and T. Goudon,
Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations, 28 (2003), 969-989.
doi: 10.1081/PDE-120021182. |
[12] |
J. A. Cañizo and H. Yoldaş,
Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, 32 (2019), 464-495.
doi: 10.1088/1361-6544/aaea9c. |
[13] |
C. Cao, The kinetic Fokker-Planck equation with weak confinement force, preprint, arXiv: 1801.10354. |
[14] |
E. A. Carlen, R. Esposito, J. L. Lebowitz, R. Marra and C. Mouhot,
Approach to the steady state in kinetic models with thermal reservoirs at different temperatures, J. Stat. Phys., 172 (2018), 522-543.
doi: 10.1007/s10955-018-2074-1. |
[15] |
P. Cattiaux and A. Guillin, Functional inequalities via Lyapunov conditions, in Optimal Transportation (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, Cambridge, 413 (2014), 274–287. |
[16] |
L. Desvillettes and C. Villani,
On the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[17] |
L. Desvillettes and C. Villani,
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[18] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[19] |
R. Douc, G. Fort and A. Guillin,
Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923.
doi: 10.1016/j.spa.2008.03.007. |
[20] |
G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate- and-fire neural network: Measure solutions and steady states, arXiv: 1710.05596. |
[21] |
W. E and D. Li,
The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.
doi: 10.1002/cpa.20198. |
[22] |
J. Evans, Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the torus, arXiv: 1702.04168. |
[23] |
N. Fournier and S. Méléard,
A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules, J. Statist. Phys., 104 (2001), 359-385.
doi: 10.1023/A:1010322130480. |
[24] |
P. Gabriel, Measure solutions to the conservative renewal equation, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 62 (2018), 68–78.
doi: 10.1051/proc/201862186206. |
[25] |
M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp. |
[26] |
M. Hairer, Lecture notes: Convergence of Markov processes, 2016. |
[27] |
M. Hairer and J C. Mattingly, Yet another look at Harrisergodic theorem for Markov chains, in Progr. Probab., (eds. Seminar on Stochastic Analysis, Random Fields and Applications VI), Birkhäuser/Springer, Basel, 63 (2011), 109–117.
doi: 10.1007/978-3-0348-0021-1_7. |
[28] |
T. E. Harris, The existence of stationary measures for certain Markov processes, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, Ⅱ (1956), 113–124. |
[29] |
F. Hérau,
Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.
doi: 10.1007/s00205-003-0276-3. |
[30] |
F. Hérau and F. Nier,
Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
|
[31] |
J. L. Lebowitz and P. G. Bergmann,
Irreversible Gibbsian ensembles, Ann. Physics, 1 (1957), 1-23.
doi: 10.1016/0003-4916(57)90002-7. |
[32] |
B. Lods and M. Mokhtar-Kharroubi,
Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.
doi: 10.1002/mma.4473. |
[33] |
B. Lods, C. Mouhot and G. Toscani,
Relaxation rate, diffusion approximation and Fick law for inelastic scattering Boltzmann models, Kinet. Relat. Models, 1 (2008), 223-248.
doi: 10.3934/krm.2008.1.223. |
[34] |
J. C. Mattingly, A. M. Stuart and D. J. Higham,
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.
doi: 10.1016/S0304-4149(02)00150-3. |
[35] |
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630.![]() ![]() ![]() |
[36] |
M. Mokhtar-Kharroubi,
On L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 226 (2014), 6418-6455.
doi: 10.1016/j.jfa.2014.03.019. |
[37] |
C. Mouhot,
Quantitative lower bounds for the full Boltzmann equation. Ⅰ. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.
doi: 10.1081/PDE-200059299. |
[38] |
C. Mouhot and L. Neumann,
Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.
doi: 10.1088/0951-7715/19/4/011. |
[39] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
show all references
References:
[1] |
D. Bakry, P. Cattiaux and A. Guillin,
Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.
doi: 10.1016/j.jfa.2007.11.002. |
[2] |
V. Bansaye, B. Cloez and P. Gabriel,
Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions, Acta Applicandae Mathematicae, (2017), 1-44.
doi: 10.1007/s10440-019-00253-5. |
[3] |
A. Bensousan, J. L. Lions and G. C. Papanicolau,
Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.
doi: 10.2977/prims/1195188427. |
[4] |
P. G. Bergman and J. L. Lebowitz,
New approach to nonequilibrium processes, Phys. Rev. (2), 99 (1955), 578-587.
doi: 10.1103/PhysRev.99.578. |
[5] |
E. Bernard and F. Salvarani,
On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954.
doi: 10.1016/j.jfa.2013.06.012. |
[6] |
M. Bisi, J. A. Cañizo and B. Lods,
Entropy dissipation estimates for the linear Boltzmann operator, J. Funct. Anal., 269 (2015), 1028-1069.
doi: 10.1016/j.jfa.2015.05.002. |
[7] |
M. Briant,
Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.
doi: 10.3934/krm.2015.8.281. |
[8] |
M. Briant,
Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.
doi: 10.1007/s00205-015-0874-x. |
[9] |
J. A. Cañizo, J. A. Carrillo and S. Cuadrado,
Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.
doi: 10.1007/s10440-012-9758-3. |
[10] |
J. A. Cañizo, A. Einav and B. Lods,
On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials, J. Math. Anal. Appl., 462 (2018), 801-839.
doi: 10.1016/j.jmaa.2017.12.052. |
[11] |
M. J. Cáceres, J. A. Carrillo and T. Goudon,
Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations, 28 (2003), 969-989.
doi: 10.1081/PDE-120021182. |
[12] |
J. A. Cañizo and H. Yoldaş,
Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, 32 (2019), 464-495.
doi: 10.1088/1361-6544/aaea9c. |
[13] |
C. Cao, The kinetic Fokker-Planck equation with weak confinement force, preprint, arXiv: 1801.10354. |
[14] |
E. A. Carlen, R. Esposito, J. L. Lebowitz, R. Marra and C. Mouhot,
Approach to the steady state in kinetic models with thermal reservoirs at different temperatures, J. Stat. Phys., 172 (2018), 522-543.
doi: 10.1007/s10955-018-2074-1. |
[15] |
P. Cattiaux and A. Guillin, Functional inequalities via Lyapunov conditions, in Optimal Transportation (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, Cambridge, 413 (2014), 274–287. |
[16] |
L. Desvillettes and C. Villani,
On the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.
doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. |
[17] |
L. Desvillettes and C. Villani,
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[18] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[19] |
R. Douc, G. Fort and A. Guillin,
Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923.
doi: 10.1016/j.spa.2008.03.007. |
[20] |
G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate- and-fire neural network: Measure solutions and steady states, arXiv: 1710.05596. |
[21] |
W. E and D. Li,
The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.
doi: 10.1002/cpa.20198. |
[22] |
J. Evans, Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the torus, arXiv: 1702.04168. |
[23] |
N. Fournier and S. Méléard,
A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules, J. Statist. Phys., 104 (2001), 359-385.
doi: 10.1023/A:1010322130480. |
[24] |
P. Gabriel, Measure solutions to the conservative renewal equation, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 62 (2018), 68–78.
doi: 10.1051/proc/201862186206. |
[25] |
M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp. |
[26] |
M. Hairer, Lecture notes: Convergence of Markov processes, 2016. |
[27] |
M. Hairer and J C. Mattingly, Yet another look at Harrisergodic theorem for Markov chains, in Progr. Probab., (eds. Seminar on Stochastic Analysis, Random Fields and Applications VI), Birkhäuser/Springer, Basel, 63 (2011), 109–117.
doi: 10.1007/978-3-0348-0021-1_7. |
[28] |
T. E. Harris, The existence of stationary measures for certain Markov processes, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, Ⅱ (1956), 113–124. |
[29] |
F. Hérau,
Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.
doi: 10.1007/s00205-003-0276-3. |
[30] |
F. Hérau and F. Nier,
Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
|
[31] |
J. L. Lebowitz and P. G. Bergmann,
Irreversible Gibbsian ensembles, Ann. Physics, 1 (1957), 1-23.
doi: 10.1016/0003-4916(57)90002-7. |
[32] |
B. Lods and M. Mokhtar-Kharroubi,
Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.
doi: 10.1002/mma.4473. |
[33] |
B. Lods, C. Mouhot and G. Toscani,
Relaxation rate, diffusion approximation and Fick law for inelastic scattering Boltzmann models, Kinet. Relat. Models, 1 (2008), 223-248.
doi: 10.3934/krm.2008.1.223. |
[34] |
J. C. Mattingly, A. M. Stuart and D. J. Higham,
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.
doi: 10.1016/S0304-4149(02)00150-3. |
[35] |
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511626630.![]() ![]() ![]() |
[36] |
M. Mokhtar-Kharroubi,
On L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 226 (2014), 6418-6455.
doi: 10.1016/j.jfa.2014.03.019. |
[37] |
C. Mouhot,
Quantitative lower bounds for the full Boltzmann equation. Ⅰ. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.
doi: 10.1081/PDE-200059299. |
[38] |
C. Mouhot and L. Neumann,
Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.
doi: 10.1088/0951-7715/19/4/011. |
[39] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
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